Coursebooks 2018-2019

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TNT: Automorphic forms and L-functions

MATH-464

Lecturer(s) :

Michel Philippe

Language:

English

Summary

Modular forms have a central place in number theory and occur also in many other branches of mathematics. Starting from the theta series associated to quadratic forms we will introduce the basic concepts associated to modular or automorphic forms and provide some applications of the theory.

Content

1-Spherical harmonics, quadratic forms and theta functions

2- The space of lattices, SL_2, the Poincaré upper-half plane and its arithmetic quotients

3-The space of holomorphic modular forms. Fourier expansion, Petersson inner product and the Petersson formula.

4-Hecke theory and L-functions

6-Applications: integral points on spheres, design of golden quantum gates, construction of Ramanujan graphs (Pizer and Lubotsky-Phillips-Sarnak)

Keywords

lattices, modular forms, L-functions, quadratic forms.

Learning Prerequisites

Required courses

Introduction to analytic number theory,

Algebraic number thery,

Analysis III,

Analysis IV

 

Learning Outcomes

By the end of the course, the student must be able to:

Teaching methods

course ex-cathedra

exercises

Expected student activities

proactive attitude during the courses and the exercises sessions (possibly with individual presentation of the solution of various problems).

 

Assessment methods

oral presentation

Supervision

Office hours No
Assistants Yes
Forum No
Others moodle page

In the programs

Reference week

 MoTuWeThFr
8-9     
9-10     
10-11MAA112    
11-12    
12-13MAA112    
13-14    
14-15     
15-16     
16-17     
17-18     
18-19     
19-20     
20-21     
21-22     
 
      Lecture
      Exercise, TP
      Project, other

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  • Autumn semester
  • Winter sessions
  • Spring semester
  • Summer sessions
  • Lecture in French
  • Lecture in English
  • Lecture in German